UC-NRLF 


*B   52fl    75^ 


1/ 


,ANE  CUBICS  WITH  A  GIVEN  QUADRANGLE 
OF  INFLEXIONS 


A  DISSERTATION 

Presented  to  the  Faculty  of  Bryn  Mawr  College  in  Partial 

Fulfillment  of  the  Requirements  for  the  Degree  of 

Doctor  of  Philosophy 


JiY 


BIRD  MARGARET  TURNER 


1920 


Reprinted  from 

American  Journal  of  Mathematics 

Vol.  XLIV,  No.  4,  October,  1922 


<v 


PLANE  CUBICS  WITH  A  GIVEN  QUADRANGLE 

OF  INFLEXIONS 


A  DISSERTATION 

Presented  to  the  Faculty  of  Bryn  Mawr  College  in  Partial 

Fulfillment  of  the  Requirements  for  the  Degree  of 

Doctor  of  Philosophy 


by 
BIRD  MARGARET  TURNER 


1920 


Reprinted  from 

American  Journal  of  Mathematics 

Vol.  XLIV,  No.  4,  October,  1922 


,e- 


V 


Reprinted  from  The  American  Journal  of  Mathematics,  Vol.  XLI,V,  No.  4,  October,  192;2 


PLANE  CUBICS  WITH  A  GIVEN  QUADRANGLE  OF  INFLEXIONS. 

By  B.  M.  Turner. 

That  every  non-singular  cubic  has  nine  points  of  inflexion,  lying  in 
related  positions  on  the  curve,  is  a  classical  fact  in  mathematics.  Of  these 
points  four  may  be  chosen  arbitrarily;  and  when  such  a  quadrangle  is 
fixed,  the  finding  of  the  positions  of  the  remaining  five  presents  a  question 
worthy  of  consideration.  It  appears  that  all  the  sets  of  five  combine  into 
a  group  of  fifteen  points  whose  relative  positions  with  respect  to  the  given 
four  depend  upon  equianharmonic  properties ;  but  that  the  equianharmonic 
relations  follow  as  a  consequence  of  a  combination  of  harmonic  relations 
and  hence,  in  a  number  of  cases,  the  points  may  be  determined  by  linear 
and  quadratic  constructions.* 

It  is  also  well  known  that  four  points  of  inflexion,  no  three  collinear, 
impose  eight  conditions  on  a  cubic  and  determine  it  as  one  of  a  singly 
infinite  system;  but,  since  only  four  of  the  conditions  are  linear  while  the 
other  four  are  of  the  third  degree,  the  system  is  not  a  pencil.  It  will  be 
shown  that  the  four  points  determine  a  system  consisting  of  six  pencils 
and  that  every  two  of  the  six  have  a  fifth  point  of  inflexion  in  common, 
that  is,  through  every  one  of  the  fifteen  points  two  of  the  pencils  pass,  and 
have  consequently  an  inflexion. 

I.  Determination  and  Construction  of  the  Remaining  Five  Inflexions. 

Four  points  of  inflexion  of  a  cubic  may  be  chosen  arbitrarily  and  the 
conditions  imposed  by  any  one  of  the  four  are  then  independent  of  the 
conditions  imposed  by  the  other  three.  For  a  real  cubic,  however,  the 
imaginary  points  of  inflexion  occur  in  conjugate  pairs;  hence  for  such  a 
cubic,  if  no  more  than  four  of  the  inflexions  are  involved  in  the  selection, 
the  chosen  quadrangle  must  consist  of  (1)  two  pairs  of  imaginary  points, 
or  (2)  two  real  points  and  one  imaginary  pair.  As  a  system  consisting 
entirely  of  imaginary  cubics  is  in  itself  of  little  interest,  only  these  two 
quadrangles  supplemented  for  symmetry  by  a  third  with  four  real  vertices 
will  be  considered  in  this  discussion.  The  results  for  the  three  cases  can 
be  stated  in  identical  terms,  as  shown  in  the  theorems  given  in  §  3  (pp. 
277-8). 

*  That  the  whole  set  of  nine  points  depends  only  on  quadratic  constructions  was 
virtually  shown  by  Mobius  (Gesammelte  Werke,  I,  p.  437)  in  the  determination  of  two 
quadrangles  in-  and  circumscribed  to  one  another.  The  eight  vertices  with  the  addition 
of  the  one  common  diagonal  point  form  the  desired  set. 

261 


520137 


'262''.        •  /  •  • :]  .Turner:  Plane  Cubics  with  Inflexions. 

j'.l.VGiiyE&.'QllABR ANGLE — TWO   PAIRS   OF   IMAGINARY   POINTS. 

Let  the  two  pairs  of  imaginary  points  to  be  taken  as  points  of  inflexion 
for  a  cubic  be  given  as  the  intersections  of  two  real  lines  with  a  conic  (Fig.  1). 


Fig.  1. 

Then,  as  is  known,  the  pencil  of  conies  through  the  four  points  has  a  real 
common  self-polar  triangle  with  A,  the  intersection  of  the  two  given  lines, 
as  a  vertex  and  BC,  the  polar  line  of  A  with  respect  to  the  given  conic,  as 
a  side.  It  is  further  known  that  the  remaining  vertices  B,  C  may  be 
determined  by  a  quadratic  construction  and  hence,  since  every  quadratic 
construction  can  be  performed  by  means  of  the  one  given  conic,  the  self- 
polar  triangle  may  be  constructed  geometrically.* 

As  the  relations  to  be  noted  are  invariant  under  projection  and  any 
four  points  forming  a  quadrangle  may  be  projected  into  the  vertices  of  any 
other  quadrangle,  there  is  no  loss  of  generality  in  choosing  the  points  as 
(i,  ±  1,  ±  l).f  Then  the  self-polar  triangle  is  the  triangle  of  reference, 
y  ±  2  =  0  are  the  given  lines,  and  the  conic  is  one  of  the  pencil 

ax2  +  by2  +  cz2  =  0 
where  —  a  +  6  +  c  =  0. 

(i)  Determination  of  the  Five  Points. 
Since  each  component  of  the  three  pairs  of  sides  of  the  quadrangle 
y*-z2=  0,        z2  +  x2  =  0,        x2  +  y2  =  0, 

passes  through  two  of  the  given  points,  the  six  lines  are  inflexional  axesj 

*  Two  pairs  of  imaginary  points  can  of  course  be  given  by  two  conies;  but  if  so  the 
determination  of  the  self-polar  triangle,  also  of  the  real  line  pair,  cannot  be  accomplished 
by  quadratic  constructions.  , 

t  Throughout  this  article  the  symbol  »  is  used  for  V  —  1  taken  positively.  In  the 
cases  where  an  ambiguity  enters,  it  is  always  preceded  by  the  double  sign. 

X  Inflexional  axis:  a  line  through  three  points  of  inflexion. 


Turner:  Plane  Cubics  with  Inflexions.  263 

for  every  cubic  inflected  at  the  four  points.  For  a  real  cubic  it  is  known 
that  the  real  sides  of  the  quadrangle  determined  by  two  pairs  of  its  imaginary 
points  of  inflexion  are  sides  of  the  real  inflexional  triangle,  the  intersection 
of  the  lines  forming  one  pair  of  imaginary  sides  of  the  quadrangle  is. a  point 
of  inflexion,  and  the  intersection  of  the  lines  forming  the  other  pair  of 
imaginary  sides  is  the  point  common  to  the  three  real  harmonic  polars. 
Hence  a  real  cubic  with  inflexions  at  the  given  four  points  has  the  lines 
through  A  as  sides  of  the  real  inflexional  triangle,  and  a  fifth  inflexion  lies 
either  at  B  or  C. 

If  the  fifth  point  of  inflexion  is  at  C,  the  third  side  of  the  real  inflexional 
triangle  passes  through  this  point  and  has  an  equation  of  the  form 
x  +  ay  —  0,  where  a  is  a  real  number  still  to  be  determined.  The  point 
B  is  common  to  the  three  real  harmonic  polars;  and  the  "line  of  reals,"* 
being  the  polar  of  B  with  respect  to  the  triangle 

y  =b  z  =  0,         x  +  ay  =  0, 

is  x  +  Say  =  0.  Hence  since  the  inflexional  axes  concurrent  with  the  real 
harmonic  polars  (z  ±  ix  =  0)  together  with  the  line  of  reals  form  a  second 
inflexional  triangle,  the  desired  cubic  is  represented  by 

(z2  +  x2)(x  +  day)  +  X(*/2  -  z2)(x  +  ay)  =  0; 

and  the  four  remaining  points  of  inflexion,  being  given  by  the  intersections 
of  x  +  Say  =  0  with  y2  —  z2  =  0  and  of  x  +  ay  =  0  with  z2  +  ar8  =  0,  are 

(3a,   —  1,  =b  1),  (a,   —  1,  ±  ia). 

These  nine  points,  namely  the  four  given  points,  the  point  C,  and  the 
four  just  found,  are  points  of  inflexion  for  a  cubic  if,  and  only  if,  they 
satisfy  the  conditions  of  collineari'ty  represented  by  the  rows,  columns, 
three  right-  and  three  left-hand  diagonals  of  the  following  scheme  :f 

(3a,  -  1,     -  1)>     ,    (    i,        1,     1),         (i,  -  1,  -  1), 

.(    *,        1,     -  1),         (3a,  -  1,     1),         (*,  -  1,        1), 

(  a,  —  1,  —  ia),         (  a,  —  1,  ia),         (0,        0,        1). 

This  requires  that  3a2  =  1.     Accordingly  the  four  points  are  either 

(V3,  -  1,  ±  1),    (1,  -  V3,  ±  t)    or    (-  V3,  -  1,  ±  1),    (1,    V3,  ±  i); 

and  the  cubic  is  a  member  of  one  jof  the  two  pencils: 


(1) 


(z2  +  z2)  (x  +  V3y)  +  \(y2  -  z2)  (x  +  ~.  y  \  -  0, 


*  The  line  through  the  three  real  points  of  inflexion. 

t  Hesse,  Crelle's  Journal  (1849),  Vol.  38,  p.  257;    also  Clebsch,  "Vorlesungen  iiber 
Geometrie,"  p.  506. 


264  Turner:  Plane  Cubics  with  Inflexions. 

(2)  (z2  +  .r2)(z  -  a/32/)  +  \{y2  -  z2)  (x  -  ±y\  -  0. 

Similarly  if  a  cubic  have  5  as  a  fifth  point  of  inflexion,  the  remaining 
four  inflexions  are  either 

(V3,  ±  1,  -  1),    (1,   =fc  i,  -  V3)    or    (-  V§,  ±  1,  -  1),    (1,  ±  i,   V§); 
and  the  corresponding  pencils  have  equations 

(3)  (y2  -  z2)(z  +  <tix)  +  \(x*+y2)(z  +  l=x\=  0, 

(4)  (f  -  z*)(z  -  V3z)  +  X(a*  +  y2)  (z  -  ^x^  =  0. 

For  symmetry,  A  is  considered  as  a  fifth  point  of  inflexion,  and  the  set 
of  nine  points  is  completed  by  either 

(±  1,   V3,  i),      (±  1,  i,  -  V3)      or      (±  1,  -  V3,  i),        (±  1,  i,   V3); 
but  the  pencils 

(5)  (*»  +  y2)(2/  +  W32)  +  X(z2  +  x2)  (y  +  ±z\  =  0, 

(6)  (z2  +  2/2)(2/  -  i  V32)  +  X(22  +  *l{v - ~«)  -  0, 

are  imaginary. 

These  results  may  be  stated  in  the  form  of  the  theorem: 

(1)  If  two  pairs  of  imaginary  points  of  inflexion  of  a  plane  cubic  are 
fixed,  a  real  fifth  point  of  inflexion  is  fixed  as  one  of  three,  and  the 
complete  group  of  nine  is  determined  as  one  of  six; 
or  in  other  words 

(1')  Two  pairs  of  imaginary  points  of  inflexion  determine  a  system 
of  cubics  consisting  of  six  syzygetic  pencils,  four  real  and  two  imaginary; 
and  every  two  of  the  six  have  a  fifth  point  of  inflexion  in  common. 

(ii)  Relative  Positions  of  the  Inflexions. 

The  curves  of  the  system  of  cubics  have  in  all  nineteen  points  of  inflexion : 
namely,  the  four  common  to  all  the  curves  and  fifteen  others  of  which  every 
one  is  common  to  two  of  the  six  pencils.  When  the  four  common  points 
are  (i,  ±  1,  db  1),  the  fifteen  are 

(    1,       0,       0),         (     0,      1,       0),         (0,       0,      1), 
^  (V3,  ±1,±  1),        (db  1,   V3,  ±  t),        (d=  1,  db  i,  V3). 

These  values  show  that  the  points  determine  certain  quadranges.    The 

lines 

y  zk  z  =  0,        zzt  ix  =  0,        x  d=  iy  =  0 


Tuener:  Plane  Cubics  with  Inflexions.  265 

are  the  sides  of  the  chosen  quadrangle;  while  the  sides  of  the  determined 
quadrangles  are 

y±         2  =  0,        z±-pz=0,        x±V§y  =  0; 
V3 

y  =fc  i  V3z  =  0,        z  ±      ix  =  0,        Z  db  -=»  =  0: 

V3 

y  zh  — =2  =  0,        8  db  V&c  =  0,        a;  ±     iy  =  0 ; 

V3 

Hence  follows  the  theorem : 

(2)  The  remaining  points  of  inflexion  of  the  cubics  with  two  common 
pairs  of  imaginary  inflexions  form  a  group  of  fifteen,  consisting  of  the 
three  diagonal  points  of  the  common  quadrangle  and  the  vertices  of 
three  other  quadrangles  with  the  same  diagonal  points,  each  of  the  three 
new  quadrangles  having  one  pair  of  sides  in  common  with  the  original 
quadrangle. 
The  sides  of  the  quadrangles  and  their  common  diagonal  triangle  are 
further  related.     The  six  real  lines  through  C  considered  as  three  pairs 

x  =  0,  y  =  0;  x  +  V3y  =  0,  x =y  =  0;  x  —  V3w  =0,  x+  -=  y  =  0 

V3  V3 

form  a  pencil  in  elliptic  involution  with 

x  ±  iy  =  0 

as  double  lines.  The  lines  of  any  one  of  the  three  pairs  are  harmonic  with 
respect  to  the  first  lines  of  the  other  two  pairs,  and  also  with  respect  to  the 
last  lines.     Furthermore  either  triad  of  lines 

x  =  0,        x  zb  V&/  =0        or        y  =  0,        x±—y=0, 

V3 

together  with  either  double  line,  forms  an  equianharmonic  system.*  Thus 
the  lines 

x±  J3y  =  0,         x±~y  =  0 
V3 

satisfy  a  combination  of  harmonic  and  equianharmonic  relations  with 
respect  to  x  —  0,  y  =  0,  x  d=  iy  =  0.  The  equianharmonic  properties, 
however,  are  simply  a  consequence  of  harmonic  properties;  for  if  we  write 

x  =  0,     y  =  0;     x  +  Py  =  0,     x  —  ay  =  0;     x  —  (3y  =  0,     x  +  ay  —  0, 

*  Consider  x  =  0,  x  *  >/3y  =  0,  x  +  iy  =  0.  Let  X  =  x  +  V%,  Y  =  x  —  V3?/; 
then  x  =  0,  x  +  iy  =  0  are  transformed  respectively  into  X  +  Y  =  0,  X  —  «F  =  0; 
and  the  cross-ratio  of  the  four  lines  X  =  0,  7  =  0,  Z  +  Y  =  0,  Z  —  uY  =  0  is  —  w2, 
where  w3  =  1.     Similarly  for  the  other  combinations. 


266  Turner  :  Plane  Cubics  ivith  Inflexions. 

and  impose  the  condition  that  x  =  0,  x  +  (3y  =  0  be  harmonic  with  respect 
to  x  —  fiy  =  0,  x  +  ay  =  0,  we  find  /3  =  3a ;  and  the  condition  that 
x  +  3a?/  =  0,  x  —  ay  =  0  be  harmonic  with  respect  to  x  ±  iy  =  0  shows 
that  3a2  =  1. 

Similar  statements  hold  for  the  lines  through  B;  and  also  for  the  set 
through  A,  except  that  in  this  case  the  involution  is  hyperbolic.  Con- 
sequently the  sides  of  the  three  quadrangles,  and  hence  the  vertices,  are 
uniquely  determined  from  the  sides  of  the  base  quadrangle  and  its  diagonal 
triangle  by  means  of  harmonic  properties. 

The  equianharmonic  properties  furnish  an  analytic  means  of  deter- 
mining the  points,  and  also  serve  to  show  the  relative  positions  of  the  three 
quadrangles  with  respect  to  the  four  given  points.  The  points  equianhar- 
monic to  (1,  0,  0)  with  respect  to  (i,  1,  1),  (i,  —  1,  1)  are  (V3,  1,  1)  and 
(3,  —  1,  —  1);  and  those  equianharmonic  with  respect  to  (i,  —  1,  1), 
(i,  1,  —  1)  are  (V3,  —  1,  1),  (V3,  1,  —  1).  Thus  the  vertices  of  the  real 
quadrangle  (\3,  ±1,  db  1)  are  the  points  on  the  lines  y  ±  z  —  0  equi- 
anharmonic to  (1,  0,  0)  with  respect  to  (i,  dz  1,  db  1).  Similarly  (dz  1, 
V3,  dz  i)  are  the  points  on  z  dz  ix  =  0  and  (1,  dz  i,  V3)  the  points  on 
x  dz  iy  =  0  equianharmonic  to  (0,  1,  0)  and  (0,  0,  1)  with  respect  to  the  four 
given  points.     These  results  are  expressed  in  the  following  theorem : 

(3)  The  fifteen  other  possible  points  of  inflexion  of  a  cubic  with 
two  fixed  imaginary  pairs  are  the  three  diagonal  points  of  the  fixed 
quadrangle,  and  the  two  points  on  every  one  of  the  six  sides  of  the 
quadrangle  equianharmonic  to  the  diagonal  point  with  respect  to  the 
two  fixed  points  on  that  side. 
The  vertices  of  the  three  determined  quadrangles  may  also  be  obtained 
analytically  as  the  intersections  of  three  conies.     Three  pairs  of  lines 

y  dz  iz  =  0,        zd=£  =  0,        xdb  y  =  0 

are  uniquely  determined  as  being  harmonic  both  with  respect  to  the  sides 
of  the  given  quadrangle  (i,  d=  1,  d=  1)  and  with  respect  to  the  sides  of  the 
diagonal  triangle.  Three  conies  having  respectively  these  pairs  of  lines  as 
tangents,  namely, 

2x*  -  y%  -  z2  =  0,        x2  -  2y2  -  z2  =  0,        x2  -  y2  -  2z2  =  0 

pass,  taken  in  the  same  order,  through  the  pairs  of  quadrangles 

(d=l,  V3,  ±i),  (±  1,  =fct,  V3); 
(dz  1,  ±h  V3),  (V3,  d=  1,  d=  1); 
(V3,  dr  1,  d=l),        (dzl,  V3,  d=i). 

Further  any  one  of  the  three  conies  passes  through  the  four  intersections 


Turner:  Plane  Cubics  with  Inflexions.  267 

of  the  pairs  of  tangents  given  for  the  other  two.  Thus  any  one  of  these 
conies  is  uniquely  determined  by  four  points  and  two  tangents,  the 
determining  elements  being  fixed  by  means  of  harmonic  properties  with 
respect  to  the  given  quadrangle.  In  turn  the  three  conies  uniquely  deter- 
mine the  three  quadrangles 

(V3,  ±1,  ±1),         (±1,  V3,  ±i),         (±1,  ±i,  V3). 

Accordingly  the  fifteen  other  possible  points  of  inflexion  of  a  cubic  with 
two  fixed  imaginary  pairs  are  the  three  diagonal  points  of  the  fixed  quad- 
rangle, and  the  twelve  intersections  of  three  conies  uniquely  determined 
analytically  by  the  quadrangle. 

(iii)  Actual  Construction  of  the  Points. 

It  has  been  noted  that  when  two  pairs  of  imaginary  points  are  given 
as  the  intersections  of  two  real  lines  with  a  conic,  there  follows  a  quadratic 
construction  for  the  triangle  self-polar  for  the  pencil  of  conies  through  the 
four  points.*  It  will  now  be  shown  that  with  the  help  of  the  triangle,  the 
fifteen  other  possible  points  of  inflexion  of  a  cubic  inflected  at  the  two  pairs 
of  imaginary  points  may  be  determined  by  a  series  of  constructions  of  which 
one  only  is  quadratic,  the  rest  linear. 

As  A  is  the  intersection  of  the  two  given  lines,  the  given  conic  meets 
BC  certainly  and  either  AB  or  CA  in  real  points.  Let  it  meet  CA  and  call 
the  points  D,  D'.  Denote  the  intersections  of  BC  with  the  given  lines 
by  E,  E'. 

The  given  points  being  («,  =fc  1,  ±  1),  the  object  is  to  construct  the  lines 

3+V3y  =  0,         «±iw  =  0,        z±V3z  =  0,        z±-Lx=0. 
V3  a/3 

The  conic  of  the  pencil  through  the  four  given  points  that  meets  y  =  0 
on  the  lines  z  ±  V3x  =  0  is  3a;2  +  by2  —  z2  =  0,  with  the  condition  that 
—  3+6—1  =  0,  that  is,  the  conic 

3z2  +  4y2  -  z2  =  0. 

This  conic  meets  x  =  0  where  4y  —  z  =  0,  hence  the  first  step  requires  the 
construction  of  the  points  of  intersection  of  the  lines  2y  —  z  =  0,  2y  +  z  =  0 
with  BC.     Since 

2y-z=0,         z=0;         y  -  z  =  0,         y  =  0, 
2y  +  z=0,         z=0;         ^+2=0,         y=0 

are  two  sets  of  harmonic  lines,  these  points  Pi,  P2  may  be  constructed 
*.See  page  262. 


268  Turner:  Plane  Cubics  with  Inflexions. 

linearly.  Draw  PJ)  intersecting  the  given  conic  in  a  second  point  D" 
and  the  given  lines  in  F,  F'.  As  the  conies  of  a  pencil  cut  any  line  in 
involution,  P3  the  conjugate  to  Pi  in  the  involution  (D"D,  FF')  is  another 
point  on  the  conic 

Sx2  +  4i/2  -  z2  =  0. 

This  conic  is  a  member  of  a  second  pencil  through  two  points  Pi  (APi 
being  a  tangent  at  a  given  point),  the  point  P2,*  and  the  point  P3.  The 
pairs  of  lines 

APlt    P2P3;        P1P2,     P1P3 

are  two  other  conies  of  the  second  pencil:  hence  this  pencil  cuts  out  on  CA 
the  involution  (AV,  CD),  where  V  is  the  intersection  of  P2P3  with  CA. 
The  involution  cut  out  on  CA  by  the  first  pencil  (the  pencil  through  the 
four  given  points)  is  defined  by  its  two  double  points,  and  may  be  expressed 
as  (A2,  C2).     It  follows  that  Q,  Q',  the  common  points  of  the  two  involutions 

(AV,  CD),        (A2,  C2), 

found  by  means  of  the  given  conic  (the  one  quadratic  construction),  are  the 
intersections  of  Sx2  -{-  4y2  —  z2  =  0  with  y  =  0.  Hence  BQ,  BQ'  are  the 
desired  lines 


db  V3z  =  0. 


Since 


z  -  -l=.x  =  0,         z  +  V3z  =  0;        z  =  0,        z  -  V3x  =  0, 
V3 

z  +  -^F.x  =  0,         z-V3z=0;        z=0,        z  +  V&r  =  0 
V3 

are  two  sets  of  harmonic  lines,  two  other  of  the  desired  lines, 

2±ia;  =  0, 
V3 

may  be  constructed  linearly:    call  them  BK,  BK'.    The  lines  that  join 
the  intersections  of  z  db  V3x  =  0,  2  zfc  —<=-  x  =  0  with  y  ±  2  =  0  to  C  are 

x±^y=  0,        x±^=y  =  0. 
V3 

Hence  the  complete  construction  (Fig.  2)  may  be  stated  as  follows: 

Construct  Pi,  P2  the  harmonic  conjugates  of  B  with  respect    to 
E,  C  and  E',  C.     Draw  PJ)  intersecting  the  given  conic  in  a  second 


*  The  conic  3x2  +  4y2  —  z2  =  0  also  has  APt  for  a  tangent,  but  the  use  of  two  points 
Pi  and  two  points  P»  would  give  an  illusory  construction. 


Turner:  Plane  Cubics  with  Inflexions. 


269 


point  D"  and  the  line-pair  in  F,  F'.  Construct  P3  the  conjugate  to 
Pi  in  the  involution  {D"D,  FF').  Draw  P2P3  intersecting  CA  in  V. 
Determine  Q,  Q'  the  common  points  *  for  the  two  involutions  (AV 


Fig.  2. 

CD),  (A2,  C2).     Construct  K,  K'  the  harmonic  conjugates  of  Q'  with 
respect  to  A,  Q  and  of  Q  with  respect  to  A,  Q'.     Draw  BK,  BK',  BQ, 
BQ'  intersecting  the  given  line-pair  in  a,  a;  (3,  (3* ';  y,  y';  5,  5'.     Draw 
a&',  a/3,  75',  y'd. 
The  seven  real  points  of  inflexion  are 

A,  B,  C,  a,  a',  13,  /3r. 


That  Q,  Q'  are  real  is  shown  by  the  analytical  discussion. 


270 


Turner:  Plane  Cvbics  with  Inflexions. 


The  eight  imaginary  points  lie  by  pairs  on  the  lines 

77',  55',  y6',  y'8, 

where  they  are  met  by  the  two  pairs  of  imaginary  sides  of  the  given 
quadrangle. 

(iv)  Symmetrical  Constructions. 
When  the  two  pairs  of  imaginary  inflexions  are  given  as  the  inter- 
sections of  two  equal  hyperbolas  with  the  same  pair  of  axes,  the  construction 
(Fig.  3)  of  the  fifteen  points  is  unique  and  furnishes  an  illustration  of  the 
three  conies  which  intersect  in  the  vertices  of  the  determined  quadrangles. 


—  *B 


Fig.  3. 

The  hypothesis  gives  the  axes  and  line  at  infinity,  that  is,  the  self-polar 
triangle;  and  (keeping  the  lettering  of  the  preceding  construction)  the  point 
A  is  the  common  center  of  the  two  hyperbolas.  Draw  the  four  other  lines 
joining  the  vertices  of  the  hyperbolas;  then  AE,  AE',  the  real  line-pair 
through  the  four  given  points,*  bisect  these  lines.     The  lines  APi,  AP2 

*  The  equality  of  the  hyperbolas  accounts  for  the  construction  of  these  lines,  in  general 
not  possible  by  quadratic  construction  when  the  two  pairs  of  imaginary  points  are  given 
by  two  conies. — See  note,  page  5. 


Turner:  Plane  Cubics  with  Inflexions.  271 

are  harmonic  to  AB  with  respect  to  AE,  CA  and  AE ' ,  CA ;  and  the  points 
Q,  Q'  are  determined  as  the  vertices  of  the  hyperbola  through  the  four  given 
points  having  AP\,  AP2  as  asymptotes.*  Then  the  lines  through  Q,  Q' 
parallel  to  AE  together  with  the  lines  joining  their  intersections  with  the 
real  line-pair  are  77',  55',  yb' ,  yf5;  and  by  means  of  the  harmonic  relations 
between  these  lines  and  the  axes  AB,  CA  the  lines  aa,  /3/3',  a/3',  a'/3  may 
be  constructed. 

The  above  construction  determines  the  vertices  of  the  three  quadrangles 
as  the  intersections  of  lines.  To  show  them  as  the  intersections  of  conies, 
let  the  two  given  hyperbolas  be  members  of  the  pencil  ax2  +  by2  +  cz2  =  0, 
where  —  a  -{-  b  -{-  c  =  0,  when  (1)  x  =  0  is  the  line  at  infinity  and  (2) 
y  =  0,  2=0  and  y  -\-  z  =  0,  y  —  2=0  are  two  pairs  of  perpendicular 
lines.  Then  the  three  conies  intersecting  in  the  vertices  of  the  determined 
quadrangles  are  two  equal,  symmetrically  placed  ellipses, 

2y2  +  22  =  x2,         y1  +  222  =  x2, 
and  the  circle 

y2  +  22  =  2.r2, 

all  three  concentric  with  the  hyperbolas. 

Accordingly  construct  the  two  equal,  symmetrically  placed  ellipses 
through  a,  a  ,  (3,  /3';  and  pass  a  circle  through  the  four  finite  intersections 
of  the  tangents  to  the  ellipses  at  their  vertices.  Then  the  fifteen  other 
possible  points  of  inflexion  of  a  cubic  inflected  at  the  four  imaginary  inter- 
sections of  the  two  hyperbolas  are  the  common  center,  the  two  points  at 
infinity  on  the  axes,  the  four  real  intersections  of  the  two  ellipses,  and  the 
eight  imaginary  intersections  of  the  circle  with  the  ellipses. 

Another  symmetrical  construction  is  obtained  by  projecting  one  pair 
of  the  given  points  into  the  circular  points.  Then,  the  pencil  of  conies 
through  the  two  pairs  of  imaginary  points  is  a  system  of  coaxial  circles,  the 
real  line-pair  consists  of  the  radical  axis  and  the  line  at  infinity,  and  two 
vertices  of  the  self-polar  triangle  are  the  limiting  points  of  the  system  while 
the  third  vertex  is  at  infinity  on  the  radical  axis.  A  pair  of  circles,  each 
having  one  limiting  point  as  a  center  and  passing  through  the  other  limiting 
point,  intersect  on  the  radical  axis  in  two  vertices  of  the  quadrangle  of  real 
points.  A  second  pair  of  circles,  having  these  vertices  as  centers  and 
passing  through  the  limiting  points,  determine  four  other  points  (2)  .on  the 
first  pair.  The  lines  joining  these  four  points  to  the  limiting  points  pass 
through  the  remaining  possible  points  of  inflexion  of  cubics  inflected  at  the 

*  Draw  a  line  parallel  to  APi  intersecting  the  real  line-pair  and  the  hyperbola  with 
AB  as  transverse  axis.  The  center  of  the  involution  determined  by  the  two  pairs  of  points 
of  intersection  is  a  point  on  the  hyperbola  having  APX  and  APi  as  asymptotes;  and  it  is 
known  that  a  hyperbola  can  be  constructed  when  the  asymptotes  and  one  point  on  the 
curve  are  given. 


272 


Turner:  Plane  Cubics  with  Inflexions. 


four  given  imaginary  points.  This  gives  a  unique  construction  when  the 
two  pairs  of  imaginary  points  are  taken  as  the  intersections  of  two  circles. 
See  (Fig.  4),  where  to  complete  the  symmetry  the  two  given  circles  are 
drawn  equal. 

y 


Fig.  4. 

For  the  proof  of  the  construction  project  (db  i,  1,  1)  into  the  circular 
points  and  change  from  homogeneous  to  Cartesian  coordinates.  The 
equation  of  the  system  of  coaxial  circles  is  then 

x2  +  y>  -  2X2/  +  X  =  0, 

with  2y  —  1  =  0  as  the  radical  axis  and  (0,  0),  (0,  1)  as  the  limiting  points. 
The  two  circles  each  having  one  limiting  point  as  center  and  passing  through 
the  other  are 

x2+y*=  1,        z2  +  (y-  1)2  =  1; 

and  these  circles  intersect  on  the  radical  axis  in  (±  £  V3,  £),  or  (zfc  V3,  1, 
—  1)  in  the  homogeneous  coordinates  given  by  z  =  y  —  1.  The  second 
pair  of  circles  having  these  two  points  as  centers  and  passing  through  the 
limiting  points,  namely, 

(z-£V3)2+  {y-\f=  1,        (*  +  £V3)2+  (y-h)2=  h 


Turner:  Plane  Cubics  with  Inflexions.  273 

determine  on  the  first  pair  the  points 

(jV3,-i),         (-*V3,  -*),         (|V3,|),         (- *V3,  I), 
or 
(1/3,  -1,-3),         (-  V3,  -  1,  -3),         (V3,3,  1),         (-V3,  3,  1); 

and  the  lines  joining  these  four  points  to  the  limiting  points  are 

a:  db  V3w  =  0,        x±-=y  =  0,        z  ±  V3s  =  0,        z±-?=.x=  0. 
V3  V3 

§  2.   Given  Quadrangle — Two  Real  and  a  Pair  of  Imaginary  Points. 

Let  four  points,  two  real  and  one  imaginary  pair,  to  be  taken  as  points 
of  inflexion  for  a  cubic  be  determined  geometrically  as  the  intersections  of 
two  real  lines  with  a  conic  (Fig.  5).     It  is  then  known  that  the  common 


Fig.  5. 

self-polar  triangle  for  the  pencil  of  conies  through  the  four  points  has  one 
real  vertex,  the  intersection  of  the  two  given  lines,  and  one  real  side,  the 
polar  of  the  real  vertex  with  respect  to  the  given  conic;  while  the  two 
remaining  vertices  and  sides  of  the  triangle  are  imaginary. 

The  study  of  the  cubics  with  two  real  and  a  pair  of  imaginary  inflexions 
fixed  is  correlated  with  the  preceding  study  of  the  cubics  with  two  fixed 
pairs  of  imaginary  inflexions,  by  choosing  the  four  points  as  (V3,  1,  ±  1), 
(1,  V3,  ±  0  * 

(i)  Determination  of  the  Five  Points. 

The  procedure  followed  in  the  case  of  the  two  pairs  of  imaginary  points 
shows  that  the  cubics  with  the  four  given  inflexions  have  six  common 
inflexional  axes,  namely,  the  three  pairs  of  side  of  the  given  quadrangle, 

x-  V&/  =  0,  x =y  =  0, 

V3 

ico2x  +  coy'rb  z  =  0,         —  icox  +  u>2y  ±  z  =  0,         (co3  =  1). 

Also  as  before  a  fifth  point  of  inflexion  is  one  of  three:   the  real  point  A 
(0,  0,  1)  or  either  of  the  pair. of  imaginary  points  B  (i,  co,  0),  C  (—i,  co2,  0). 
*  See  page  263. 


274  Turner:  Plane  Cubics  ivith  Inflexions. 

Consider  first  a  cubic  with  the  real  point  A  as  a  fifth  point  of  inflexion. 
The  cubic  has  two  imaginary  inflexional  axes  through  this  point;  and  the 
equation  is  consequently  of  the  form 

(iuPx  +  cot/  +  z)  (ico2x  +  <ay  —  z)(x  +  ay) 

+  X(—  icox  -\-  u?y  +  z)(—  i(»x  +  oo2y  —  z)(x  +  fiy)  =  0, 

where  a,  /3  are  a  pair  of  complex  numbers.  Accordingly  the  remaining  four 
inflexions  are  at 

(a,  —  1,  iua  +  co2),         (a,  —  1,  —  icoa;  —  co2), 
(0,  -  1,  ioflS  -  co),         (ft  -  1,  -  irf(3  +  co), 

where,  in  order  to  satisfy  the  conditions  of  collinearity  imposed  on  every 
group  of  inflexions  of  a  non-singular  cubic,  that  is,  the  conditions  repre- 
sented by  the  scheme 

( V3,  1,  1),         (0,  -  1,  ~  u/0  +  co),         (a,  -  1,  iua  +  co2), 
08,  -  1,  ico2/?  -_  co),         ( V3,  1,  -  1),         {a,  -  1,  -  icoa  -  co2), 
(1,  V3,  0,         (1,  V3,  -  i),         (0,  0,  1) 

either  a  =  -  i,  |8  =  i,  or  a  =  }(-  i  *-  4  V3),  0=  |(t  —  4  V3). 

If  a  =  —  »,  j8  =  i,  the  four  points  are  (»,  ±  1,  ±  1)  in  agreement  with 
the  results  in  the  preceding  case.  If  a  ==  y(—  z  —  4a/3),  |8  =  j(i  —  4^3), 
a  second  set  of  four  points  is  obtained;  but  since  the  computations  involved 
are  complicated  it  is  advantageous  to  apply  a  linear  transformation  by 
which  the  original  four  points  become 

(0,  1,  -  1),         (-  1,  0,  1),         (1,  -  co,     0),         (1,  -  co2,  0). 
Then  the  fifth  point  is  (1,  —  1,  0);  the  remaining  four  are  either 

(0,  1,  -  co),         (0,  1,  -  co2),         (-  co,  0,  1),         (-  co2,  0,  1), 

or 

(-  1,  co2  -  1,  1),    (-  1,  co  -  1,  1),    (co2  -1,-1,  1),    (co  -  1,  -  1,  1): 

and  the  corresponding  pencils  of  cubics  may  be  written  as 

3?  +  V3  +  z3  +  ^xyz  =  0, 
x3  +  y3  +  z3  +  3z(^  +  y2)  +  Zz2(x  +y)  +  \z(z  +  x)(y  +  z)  =  0. 

Similarly  if  either  B  or  C  is  the  fifth  point  of  inflexion,  there  are  two 
distinct  pencils  of  cubics;  but,  since  the  four  pencils  thus  determined  are 
imaginary  and  consequently  of  interest  in  this  discussion  only  to  give  sym- 
metry to  the  results,  their  equations  and  the  coordinates  of  their  remaining 
points  of  inflexion  are  omitted. 


Turner:  Plane  Cubics  with  Inflexions.  275 

The  cubics  of  the  three  pairs  of  pencils  have  in  all  only  nineteen  points 
of  inflexion,  and  these  have  the  same  relative  positions  as  the  nineteen  for 
the  three  pairs  of  pencils  determined  by  two  pairs  of  imaginary  inflexions. 
Hence,  it  follows  that,  with  the  proper  interchanging  of  the  words  real  and 
imaginary,  the  theorems  stated  on  pages  264,  265,  and  266  hold  for  the  cubics 
with  two  real  and  an  imaginary  pair  of  fixed  inflexions.  The  constructions, 
however,  because  of  the  great  number  of  imaginary  elements  involved 
become  almost  entirely  theoretical.  ,      , 

(ii)  Related  Quartic  Curves. 
The  equations 

-Pi  =  x3  -f-  y3  +  z3  +  \xyz  =  0, 

P2  =  x3  +  y3  +  zz  +  Sz(x2  +  y2)  +  3z\x  +  y)  +  \z(z  +  x)  (x  +  y)  =  0 

represent  two  pencils  of  cubics  having  in  common  three  real  and  a  pair  of 
imaginary  inflexions.  Every  value  of  X  determines  a  definite  curve  in  each 
pencil,  and  the  elimination  of  X  between  the  two  equations  gives 

z'[(x3  +  y3  +  z3){x  +  y  +  z)  -  Sxy(x2  +  f)  -  Sxyz(x  +  y)]  -  0 

as  the  locus  of  the  intersections  of  the  two  curves.  Three  of  the  fixed  inter- 
sections, namely,  one  real  and  the  given  pair  of  imaginary  inflexions,  are  on 
s=0;  hence  this  line  is  a  part  of  the  locus  only  because  of  the  two  curves 
of  which  it  forms  a  part.  The  remaining  two  fixed  inflexions  and  the  four 
variable  intersections  lie  on  the  quartic 

(x3  +  y3  +  z3)(x  +  y  +  z)  —  3xy(x2  +  y2)  —  3xyz(x  +  y)  =  0. 

The  remaining  two  fixed  inflexions  are  (0,  1,  —  1),  (—  1,  0,  1),  where  the 
quartic  has  nodes  with  tangents 

=b  ix  +  y  +  2  =  0,        x  ±  iy  +  z  =  0; 

and  these  lines  are  the  inflexional  tangents  common  to  the  two  cubics 
considered  when  X  =  ±  Si.  Thus  the  binodal  quartic  is  the  locus  of  the 
four  variable  interesections  of  the  two  curves  of  Pi  and  P2  having  the  same 
parametric  value. 

The  two  sets  of  four  points 

(0,  1,  -  co),         (0,  1,  -  co2),         (-  co,  0,  1),  ;       (-  co2,  0,  1); 
(-  1,  co2  -  1,  1),    (-  1,  co  -  1,  1),     (co2  -1,-1,  1),     (co  -1,-1,  J), 

completing  the  inflexional  groups  of  Pi  and  P2  are  on  the  quartic ;  and  these 
together  with  the  two  double  points  (0,  1,  —  1),  (—  1,  0,  1)  are  the  complete 
intersection  of  two  quartics 

(wx  +  y  +  z)  (u2x  +  y  +  2)  (x  +  wy  +  z)  (x  +  u2y  +  z)  =  0, 
xy(z+  x)(x+  y)  =  0, 


276  Turner:  Plane  Cubics  with  Inflexions. 

each  composed  of  two  pairs  of  lines,  one  pair  through  each  of  the  double 
points.     Accordingly 

(<ax  +  y  +  z)(u?x  +  y  +  z)(x  +  coy  +  z)(x  +  w2y  +  z) 

+  nxy(z  +  x)(x  +  y)  =  0 

is  a  pencil  of  binodal  quartics  through  the  eight  points  and  two  nodes.  If 
the  second  pencil  of  cubics  is  written 

P2  m  x*+  tf+  z3  +  Sz(x*+  2/2)  +  3z2(z  +  y)  +  (X-  n)z(z+  x)(y  +  z)  =  0, 

where  n  has  any  real  value,  a  different  quartic  of  the  pencil  corresponds  to 
each  chosen  value  of  n,  that  is,  the  manner  of  writing  the  equations  of  Pi 
and  P2  can  be  so  varied  that  every  quartic  of  the  pencil  may  be  found  as 
the  locus  of  the  four  variable  intersections  of  the  two  cubics  with  the  same 
parametric  value.  The  two  reducible  quartics  corresponding  to  n  infinite 
or  zero  are  obtained  respectively  when  n  =  <x>  or  n  =  3.  If  n  =  <x> ,  P2 
breaks  up  into  three  lines.  If  n  =  3,  the  two  cubics  have  the  same  tangents 
at  the  two  common  imaginary  inflexions  (1,  —  co,  0),  (1,  —  co2,  0)  for  every 
value  of  X;  and  all  their  intersections  lie  at  the  five  given  points  except 
when  the  cubics  have  a  common  linear  factor.  In  the  study  of  non-singular 
cubics  these  two  cases  are  excluded,  and  hence  the  following  result  can  be 
stated : 

There  are  two,  and  only  two,  pencils  of  cubics  having  in  common  three 
real  and  a  pair  of  imaginary  inflexions;  and  the  locus  of  the  four  variable 
intersections  of  the  two  corresponding  curves  is  a  binodal  quartic  passing 
through  the  remaining  inflexions  of  the  two  pencils,  the  two  nodes  being 
at  the  two  real  inflexions  not  collinear  with  the  two  common  imaginary 
inflexions. 

In  further  consideration  of  the  geometry  on  the  pairs  of  cubics  and  the 
resulting  quartic  curve  it  may  be  noted  that  three  of  the  nine  intersections 
of  the  two  cubics  lie  on  a  line,  hence  the  remaining  six,  the  six  on  the  quartic, 
lie  on  a  conic.     The  equation 

Pi  -  P2  =  3z[>2  +  f  +  z(x  +  y)  +  \z{x  +  y  +  z)  =  0 

represents  a  pencil  of  cubics  consisting  of  the  line  and  a  pencil  of  conies. 
Every  value  of  X  determines  a  curve  of  Pi  and  P2  and  a  conic  through  their 
six  intersections  on  the  quartic.  The  pencil  of  conies  passes  through  the 
two  nodes  (0,  1,  —  1),  (—  1,  0,  1),  and  the  two  points  (1,  ±  i,  0)  where  the 
line  z  =  0  is  intersected  by  the  nodal  tangents 

±  ix  +  y  +  z  =  0,        x  db  iy  +  z  =  0. 

Thus  the  four  variable  intersections  of  the  two  cubics  are  cut  out  on  the 


Turner:  Plane  Cubics  with  Inflexions.  277 

binodal  quartic  by  a  pencil  of  concis  through  the  two  nodes  and  the  two 
intersections  of  the  nodal  tangents  collinear  with  the  two  common  imaginary 
inflexions. 

When  the  points  (1,  ±  i,  0)  are  projected  into  the  circular  points,  the 
special  case  arises  where  the  four  variable  intersections  lie  on  a  circle  through 
two  of  the  common  real  points  of  inflexion.  It  may  also  be  noticed  that, 
since  the  four  variable  intersections  of  the  two  cubics  and  the  two  nodes 
of  the  quartic  lie  on  a  conic,  the  four  variable  intersections  subtend  at  the 
two  nodal  points  pencils  of  lines  with  the  same  cross-ratio. 

§  3.   General  Conclusions. 

For  symmetry  the  cubics  with  a  fixed  quadrangle  of  real  points  of  inflex- 
ion are  considered,  although  every  such  cubic  is  imaginary.  Choose  the 
four  points  ( Af3,  =fc  1,  ±  1).     Then  the  six  fixed  inflexional  axes  are 

y±z=0,        2±is  =  0,        z±V3i/  =  0; 
V3 

and  a  fifth  point  of  inflexion  is  any  one  of  the  three 

(1,  0,  0),         (0,  1,  0),         (0,  0,  1). 

It  follows  that  a  cubic  of  the  system  with  an  inflexion  at  (1,  0,  0)  is  a  member 
of  one  of  the  pencils 

(z2  -  3y*)(y  +  i  V3z)  +  Ms2  -  i*2)  (y  +  ~«)  =  0, 

(z2  -  3y*)(y  -  i&)  +  \(z2  -  J>)  (y  -  ±z\  =  0; 

and  similar  results  hold  with  respect  to  (0,  1,  0)  and  (0,  0,  1).  Furthermore 
the  six  pencils  have  nineteen  points  of  inflexion  associated  as  in  the  two 
preceding  cases.  Hence  the  theorems  already  stated  (pp.  264,  265,  266)  are 
applicable  to  this  case  also,  that  is,  for  cubics  with  any  fixed  quadrangle  of 
inflexions*  the  following  theorems  hold: 

f  (1)  If  four  points  of  inflexion  of  a  plane  cubic,  no  three  collinear,  are 
fixed,  a  fifth  point  of  inflexion  is  fixed  as  one  of  three,  and  the  complete 
group  of  nine  is  determined  as  one  of  six;  or  in  other  words, 

(1 )  A  quadrangle  of  inflexions  determines  a  system  of  cubics  consisting 
of  six  syzygetic  pencils,  and  every  two  of  the  six  have  a  fifth  point  of  inflexion 
in  common. 


*  Provided,  as  stated  on  page  261,  that  if  imaginary  the  points  enter  by  conjugate  pairs. 
t  A.  Wiman,  Nyt  Tiddskrift  for  Matematic  (1894);   also  W.  Burnside,  Proc.  London 
Math.  Soc.  (1906-07). 


278  Turner:  Plane  Cubics  with  Inflexions. 

(2)  The  remaining  points  of  inflexion  of  the  cubics  with  a  common 
quadrangle  of  inflexions  form  a  group  of  fifteen,  consisting  of  the  three 
diagonal  points  of  the  common  quadrangle  and  the  vertices  of  three  other 
quadrangles  with  the  same  diagonal  points,  each  of  the  three  new  quad- 
rangles having  one  pair  of  sides  in  common  with  the  original  quadrangle. 

(3)  The  fifteen  other  possible  points  of  inflexion  of  a  cubic  with  a  fixed 
quadrangle  of  inflexions  are  the  three  diagonal  points  of  the  fixed  quadrangle, 
and  the  two  points  on  every  one  of  the  six  sides  of  the  quadrangle  equi- 
anharmonic  to  the  diagonal  point  with  respect  to  the  two  fixed  points  on 
that  side. 


VITA 

I,  Bird  Margaret  Turner,  was  born  in  Moundsville,  West  Virginia,  April  18,  1877.  My  father 
was  John  Marion  Turner  and  my  mother  Mary  J.  (Douglas)  Turner.  In  1893  I  was  graduated 
from  the  Moundsville  High  School,  in  1915  received  the  degree  of  Bachelor  of  Arts  and  in  1917 
the  degree  of  Master  of  Arts  from  the  West  Virginia  University. 

From  1900  to  1913  I  was  teacher  of  mathematics  and  in  1915-16  Principal  of  the  Mounds- 
ville High  School;  from  1913  to  1915  I  was  Student  Assistant  in  Mathematics  at  the  West  Vir- 
ginia University.  During  the  year  1917-18  I  was  Assistant  Director  of  the  Phebe  Anna  Thorn 
Model  School  and  in  1918-19  part  time  Reader  in  Mathematics  at  Bryn  Mawr  College. 

I  entered  Bryn  Mawr  College  as  Scholar  in  Mathematics  in  1916,  was  granted  the  President 
M.  Carey  Thomas  European  Fellowship  for  my  first  year's  work,  and  was  Resident  Fellow  in 
1919-20. 

At  Bryn  Mawr  College  I  studied  under  the  direction  of  Professors  Charlotte  A.  Scott,  Anna 
J.  Pell,  Matilde  Castro  and  Dr.  Olive  Clio  Hazlett;  at  the  West  Virginia  University  under  Pro- 
fessors John  Arndt  Eiesland  and  Joseph  Ellis  Hodgson.  It  gives  me  great  pleasure  to  have  this 
opportunity  of  expressing  to  all  these  professors  my  gratitude  for  their  valuable  instruction.  In 
particular  my  thanks  are  extended  to  Professor  C.  A.  Scott  for  her  constant  encouragement 
during  my  connection  with  Bryn  Mawr  College,  and  for  her  helpful  criticism  and  unfailing  inter- 
est throughout  the  preparation  of  this  dissertation. 


■P  '*■"--  " 


TfflS  «^-2S^    cENTS 

triME       °F  ,«   RETURN 

am    INITIAL    F1^bFm^^tHte°pSauty 

AN       Jp    ASSESSED    FOR  ue-      THEP  TH 

oVERDuE- 


Photomount 

Pamphlet 

Binder 

Gaylord  Bros. 

Makers 
Syracuse,  N.  Y. 

PAT.  JAN  21, 1908 


.-■:■'■*- 


•  ■•,,■.;■••-■■    w 

.'■..-. 


-■■'••'-'■■•'.":■'. 


520  i  3 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


WMmmm 

alSglEOfsalliSl ! 


